*Time-Asymptotic Interactions of Boltzmann* *Shock Layers in the Presence of Boundary*

### W

EN### -C

HING### L

IEN*&* S

HIH### -H

SIEN### Y

UABSTRACT. We study the time-asymptotic behavior of the Boltzmann
shock layers with a given physical boundary in a half-space. As boundary
conditions, we prescribe a Maxwellian at the far field and require a spec-
ular reflection at the wall*x**=*0. When the macroscopic velocity at the
far field is negative, we prove that if the initial data are suitably chosen,
then a solution exists globally in time and tends toward the correspond-
ing outgoing Boltzmann shock profile as time goes to infinity. The proof
is essentially based on the macro-micro decomposition of solutions and
the elementary energy methods.

CONTENTS

1. Introduction. . . 1502

2. Preliminary. . . 1507

2.1. Macro-micro decomposition.. . . 1507

2.2. Boltzmann shock profile.. . . 1510

3. Construction of the Approximate Solution. . . 1512

3.1. The macroscopic and microscopic equations.. . . 1512

3.2. Reference macro-micro decomposition and reference norms.. 1516

3.3. Local existence in time.. . . 1517

4. Basic Estimates. . . 1518

4.1. Properties of the macroscopic variables.. . . 1518

4.2. Matrix representation.. . . 1519

4.3. The eigenvalues for the Navier-Stokes shock profile.. . . 1520

5. Stability Analysis. . . 1521

5.1. Lower order energy estimates.. . . 1524

5.2. Transversal wave estimates.. . . 1533 5.3. Higher order energy estimates.. . . 1541 App. A. Chapman-Enskog Expansion andNavier-Stokes Shock Profiles. 1549 App. B. Estimates on Collision Operators. . . 1553 Acknowledgment.. . . 1554 References. . . 1554

1. INTRODUCTION

Consider the Boltzmann equation with a given physical boundary in a half space.

We study the plane wave propagation on the half space*R*^{+}*= [*0*,**∞)* with the
following initial and boundary conditions:

*F**t**+ ξ*1*F**x* *= Q(F, F),* *(x, t, ξ)**∈ R*^{+}*× R*^{+}*× R*^{3}*,*
(1.1)

*x→∞*lim*F (x,*0*, ξ)**= ρ**+**e*^{−|ξ−u}^{+}^{|}^{2}^{/(}^{2}^{T}^{+}* ^{)}*
p

*(*2

*π T*

_{+}*)*

^{3}

*,*(1.2)

*F (*0*, t, ξ*_{1}*, ξ*_{2}*, ξ*_{3}*)**= F(*0*, t,**−ξ*1*, ξ*_{2}*, ξ*_{3}*),*
(1.3)

where*F (x, t, ξ)*is the density function of the gas at time*t**≥*0, position*x* *≥*0,
and with the velocity *ξ* *= (ξ*1*, ξ*2*, ξ*3*)*. *ρ** _{+}*,

*u*

_{+}*= (u*

*+*1

*,*0

*,*0

*)*, and

*T*

*are the macroscopic density, the macroscopic velocity, and the temperature respectively in the thermal equilibrium at the far field. We assume that*

_{+}*u*

*1*

_{+}*<*0. Here the specific gas constant is assumed to be one.

We also assume the hard sphere collision:

*Q(g, h)**≡*
Z

*R*^{3}

Z

*S*^{2}

h

*g(ξ*^{0}*)h(ξ*_{?}^{0}*)**− g(ξ)h(ξ**?**)*i

*C(Ω, ξ − ξ**?**)*^{d}Ω^{d}*ξ**?**,*

where*C(**Ω, ξ − ξ**?**)**≡ |Ω · (ξ − ξ**?**)**|*, and

*ξ*^{0}*= ξ + (Ω · (ξ**?**− ξ))Ω, ξ**?*^{0}*= ξ**?**− (Ω · (ξ**?**− ξ))Ω,* *Ω ∈ S*^{2}*.*
In this paper we are interested in the time-asymptotic behavior of the Boltz-
mann shock layers. According to condition (1.2), the incoming flow travels along
the positive*x*-axis and strikes the boundary*x* *=* 0. Due to condition (1.3), the
incoming flow will be reflected as a wave of some form. We expect that the outgo-
ing flow will tend toward a Boltzmann shock profile as time goes to infinity, which
is a solution of the equation

*−sψ*^{0}*+ ξ*1*ψ*^{0}*= Q(ψ, ψ).*

Here*s*is the speed of the shock wave, and the space variable is one dimensional,
*x**∈ R*. Motivated by the boundary condition, we first construct the approximate
solution*ϕ*of (1.1)–(1.3) by superposing two travelling shock wave solutions mov-
ing in the opposite directions with the same speed. By writing the solution of (1.1)
as*F (x, t, ξ)**= ϕ(x, t, ξ) + J(x, t, ξ)*, we obtain the equation for*J*as follows:

(1.4) *J**t**+ ξ*1*J**x**= Q(ϕ + J, ϕ + J) − Q(ϕ, ϕ) − (ϕ**t**+ ξ*1*ϕ**x**− Q(ϕ, ϕ)).*

Our goal is to prove the time-asymptotic convergence of the solution to (1.1)–

(1.3) toward the outgoing Boltzmann shock profile; that is, *J(x, t, ξ)* decays to
zero in a suitable norm as*t**→ ∞*.

There is extensive literature on the initial boundary value problems for the
Boltzmann equation initiated by Cercignani. Existence, uniqueness and proper-
ties of asymptotic behavior are proved for solutions of the Milne and Kramers
problems, which are to solve the linearized Boltzmann equation in a half space
*x >*0. Such studies on stationary solutions have been pursued analytically by [2],
[8], [9], [27]. The Milne problem has also been studied by asymptotic expansions
for the condensation and evaporation by Sone et al [1], [23].

The shock profiles of the Boltzmann equation are first constructed by [4], where the existence and uniqueness of weak plane shocks are obtained by using a projection method similar to Lyapunov-Schmidt method; however, the positivity property of the Boltzmann shock profile cannot be concluded from this approach.

The time-asymptotic stability of the Maxwellian states has been shown by energy methods based on the Fourier transform and spectral analysis [15,22,24,25].

Furthermore, the time-asymptotic stability and positivity of shock profiles are ob- tained in [18]. An elementary energy method is introduced in [18] based on a macro-micro decomposition of the equation into macroscopic and microscopic components to analyze the time-asymptotic stability of nonlinear waves. The de- composition effectively describes the Boltzmann dynamics so that the methods of analyzing viscous conservation laws can be implemented with small modifications.

The positivity of a Boltzmann shock profile is thus shown by the time-asymptotic approach and the maximal principle for the collision operator.

Applying the macro-micro decomposition introduced in [18], we can regard the present problem as a time-asymptotic stability problem, of which the solution tends toward the nonlinear wave pattern, a superposition of two Boltzmann shock layers. We briefly describe the key steps for solving this problem in the following.

**Step 1. We first construct the approximate solution***ϕ*of (1.1)–(1.3). (See
Sections2.2and3.1).

The state*(ρ*_{+}*, m*_{+}*,*^{E}_{+}*)*is given by (1.2) at*x* *= ∞*. Since (1.3) implies that
the macroscopic velocity and momentum are zero at*x* *=* 0, we can construct a
Boltzmann shock profile with these two given conditions. By solving the Rankine-
Hugoniot condition (2.4) and equation (2.3), there exist*ρ*_{0},^{E}0and a wave speed
*s >* 0 such that *(ρ*_{+}*, m*_{+}*,*^{E}_{+}*)*and *(ρ*_{0}*,*0*,*^{E}_{0}*)*can be connected by a travelling

wave solution*ϕ*_{+}*(x**− st, ξ)*on the whole space*R**= (−∞, ∞)*satisfying

*−sϕ*^{0}_{+}*+ ξ*1*ϕ*^{0}_{+}*= Q(ϕ**+**, ϕ*_{+}*).*

Let*(ρ*_{+}*(x**−st), m**+**(x**−st),*^{E}*+**(x**−st))*denote the corresponding macroscopic
variables of*ϕ*_{+}*(x**− st, ξ)*. We then construct*ϕ*_{−}*(x, ξ*_{1}*, ξ*_{2}*, ξ*_{3}*)**≡*

*ϕ*_{+}*(**−x, −ξ*1*, ξ*_{2}*, ξ*_{3}*)*. We now choose the approximate solution*ϕ(x, t, ξ)*to be
*ϕ(x, t, ξ)**≡ ϕ**+**(x**− s(t + t*0*), ξ)**+ ϕ**−**(x**+ s(t + t*0*), ξ)**− ρ*0*ω(ξ*; 0*, T*_{0}*),*
where

*ω(ξ*; 0*, T*_{0}*)**=* *e*^{−|ξ|}^{2}^{/(}^{2}^{T}^{0}* ^{)}*
p

*(*2

*π T*0

*)*

^{3}

*.*

Here*ε* *≡ |ρ**+**− ρ*0*| *1 and *T*0is the temperature at *x* *=* 0. We choose*t*0 *≡*
*ε*^{−}^{3} large enough to study the time-asymptotic state. It follows from the above
construction that

*ϕ(x, t, ξ*_{1}*, ξ*_{2}*, ξ*_{3}*)**= ϕ(−x, t, −ξ*1*, ξ*_{2}*, ξ*_{3}*).*

**Step 2. Focus on the following equation and choose the suitable initial state***J(x,*0*, ξ)*. (See Sections3.1and3.3).

*J**t**+ ξ*1*J**x* *= Q(ϕ + J, ϕ + J) − Q(ϕ, ϕ) − (ϕ**t**+ ξ*1*ϕ**x**− Q(ϕ, ϕ)).*

Let *F* denote a solution of (1.1)–(1.3), extended to the whole space *R* by
setting

*F (x, t, ξ*_{1}*, ξ*_{2}*, ξ*_{3}*)**= F(−x, t, −ξ*1*, ξ*_{2}*, ξ*_{3}*),* for*x <*0*.*
We treat*F* as a perturbation of the approximate solution*ϕ*. Thus, we write

*F (x, t, ξ)**= ϕ(x, t, ξ) + J(x, t, ξ).*

Therefore,*J(x, t, ξ)*satisfies the above equation (1.4). We then choose the initial
state*J(x,*0*, ξ)*satisfying

Z_{∞}

*−∞*

Z

*R*^{3}

1
*ξ**i*

*|ξ|*^{2}

* J(x,*0*, ξ)*^{d}*ξ*^{d}*x**=*0*,* for*i**=*1*,*2*,*3*.*
(1.5)

Due to the conservation laws for the macroscopic variables, it follows that
Z_{∞}

*−∞*

Z

*R*^{3}

1
*ξ**i*

*|ξ|*^{2}

* J(x, t, ξ)*^{d}*ξ*^{d}*x**=*0*,* for*i**=*1*,*2*,*3*.*

Such property on macroscopic variables allows one to introduce an anti-derivative variable and the energy method can be applied to study the time-asymptotic sta- bility problem.

* Step 3. Derive the equations for applying the energy method. (See Section*
3.1for details.)

Introduce the anti-derivative:

*W (x, t, ξ)**≡*
Z_{x}

*−∞**J(y, t, ξ)*^{d}*y.*

We obtain from (1.4)
(1.6) *W**t**+ ξ*1*W**x* *=*

Z_{x}

*−∞**(Q(ϕ**+ J, ϕ + J) − Q(ϕ, ϕ) − E(ϕ))*^{d}*y,*
where*E(ϕ)**= ϕ**t**+ ξ*1*ϕ**x**− Q(ϕ, ϕ)*.

We need to make use of the shock profile of the Navier-Stokes equation. Let
*u*NS and *T*NS denote the velocity and temperature for the corresponding shock
profile of the Navier-Stokes equation constructed by the same conditions imposed
on*ϕ*at the far field. We denote the corresponding local Maxwellians by

*ω*tr*(x, t, ξ)**=* *e*^{−((ξ}^{1}^{−u}^{NS}^{)}^{2}^{+ξ}^{2}^{2}^{+ξ}^{3}^{2}^{)/(}^{2}^{T}^{NS}* ^{)}*
p

*(*2

*π T*NS

*)*

^{3}

and the collision invariants*ψ**i**(x, t, ξ)*,*i**=*0*, . . . ,*4, are as follows:

*ψ*0*(x, t, ξ)**=*1*,*
*ψ*_{1}*(x, t, ξ)**=* *ξ*1p*− u*^{NS}

*T*NS

*,*
*ψ**i**(x, t, ξ)**=* p*ξ**i*

*T*NS

*,* *i**=*2*,*3*,*
*ψ*4*(x, t, ξ)**=* * _{√}*1

6

*(ξ*_{1}*− u*^{NS}*)*^{2}*+ ξ*_{2}^{2}*+ ξ*_{3}^{2}
*T*NS

*−*3

!
*.*

We introduce the macroscopic and microscopic variables*W*_{0}and*W*_{1}for*W*:

*W*_{0}* ≡ P*0

*W*

*≡*X4

*i*

*=*0

Z

*W ψ*_{i}^{d}*ξ*

*ψ*_{i}*ω*tr*,*
*W*1* ≡ P*1

*W*

*≡ W − W*0

*,*

where**P**_{0}is the projection operator on the space spanned by*ψ**i**ω*tr,*i**=*0*, . . . ,*4
and**P**_{1}is the orthogonal projection**P**_{1}* = I − P*0. We also decompose

*J*as

*J**= J*0*+ J*1*,* *J*_{0}* ≡ P*0

*J,*

*J*

_{1}

*1*

**≡ P***J.*

Applying**P**_{0}to equation (1.6) and**P**_{1}to equation (1.4) separately, we obtain
the desired equations as follows:

(*†*) **P**_{0}*∂**t**W*_{0}* + P*0

*ξ*

_{1}

**P**

_{0}

*∂*

*x*

*W*

_{0}

*0*

**+ P***ξ*

_{1}

*J*

_{1}

*=*0.

(*††*) **P**_{1}*∂**t**J*_{0}* + P*1

*∂*

*t*

*J*

_{1}

*1*

**+ P***ξ*

_{1}

*∂*

*x*

*J*

_{0}

*1*

**+ P***ξ*

_{1}

*∂*

*x*

*J*

_{1}

*− L(J*1

*)*

*= D(J) +*

^{N}

*(J)*

*− E(ϕ)*. It should be noticed that we only need the component

*W*

_{0}and

*J*in the proof, and the local existence in time of

*W*

_{0}and

*J*is shown in Section3.3.

We note that the projection**P**_{0} to equation (1.6) can be regarded as a lin-
earization around the Navier-Stokes shock profile in the sense that the physical
quantities*u*and*T* in the background Maxwellian are chosen from the shock pro-
file of the Navier-Stokes equation.

**Step 4. Apply the elementary energy method to***(**†)*and*(**††)*to prove that
the solution*J(x, t, ξ)*decays to zero in the*k · k*ref*,L*^{∞}_{x}*(L*^{2}_{ξ}*)* norm as*t* *→ ∞*. Here,
the reference norm is defined by

*khk*ref*,L*^{∞}*x**(L*^{2}_{ξ}*)**=*sup

*x∈R*

Z

*R*^{3}

*h*^{2}*(*2*π T*_{0}*)*^{3}^{/}^{2}*e*^{|ξ|}^{2}^{/(}^{2}^{T}^{0}^{)}^{d}*ξ.*

(See Section5for details.)

We share several technical difficulties with [18]. For the sake of complete- ness, we address them again. First, we need to apply the transversal wave estimate.

When we apply the theory of hyperbolic conservation laws to the macroscopic
part, we recover the 3 families of elementary waves. In the present problem, the
compressibility of the Navier-Stokes shock profile can be used in the estimate in-
volving the first and third families, but not for the second family which produces
transversal terms. Originated from the energy estimate for the stability analysis
of a viscous shock profile [10], we refine the transversal wave estimate in the cur-
rent situation. Secondly, the Boltzmann equation is nonlinear due to the collision
operator. We split the collision operator*Q*into the linear part*L*and the nonlin-
ear part*N*. The negative definiteness of*L*yields the decaying of the microscopic
component. But we are left with the nonlinear part*N*, which produces terms like
*k(*1*+|ξ|)*^{1}^{/}^{2}*J*1*k**L*^{2}* _{ξ}*. We therefore introduce the norms:

*k·k*ref

*,L*

^{∞}

_{x}*(L*

^{2}

_{ξ}*)*,

*k·k*ref

*,L*

^{∞}

_{x,t}*(L*

^{2}

_{ξ}*)*. (See Section3.2.) It leads us to make the right a priori assumption and establish the higher order energy estimate to resolve the nonlinear terms. Finally, we have error terms caused by the approximate solution and the drifting Maxwellians. The facts that the projection

**P**0is determined by the physical quantities of the Navier- Stokes shock profile and the local Maxwellians

*ω*tr vary along the same shock profile certainly result in several error terms. When the shock strength is suffi- ciently small, the Boltzmann shock and the Navier-Stokes shock are close enough, which allows us to control all those errors. In addition, Kawashima’s method [13]

is applied to control the density term, which is absent in considering the pertur- bation of a global Maxwellian.

We state the main theorem as follows:

**Theorem 1.1. Consider the hard sphere model of equations (1.1)–(1.3). Suppose***that the shock strength of**ϕ**is sufficiently small. Under the condition (*1.5), there exists

*a constant**δ*_{0}*>**0 such that if the initial data are sufficient small:*

X

*|α|≤*4

*k∂**x*^{α}*W*_{0}*k**L*^{∞}_{x,t}*(L*^{2}_{ξ}*)**+ k∂**x*^{α}*∂**t**W*_{0}*k**L*^{∞}_{x,t}*(L*^{2}_{ξ}*)*

*+* X

*|α|≤*3

*k∂**x*^{α}*{(*1*+ |ξ|)*^{1}^{/}^{2}*J*1*}k*ref*,L*^{∞}_{x,t}*(L*^{2}_{ξ}*)*

*+ k∂*^{α}*x**∂**t**{(*1*+ |ξ|)*^{1}^{/}^{2}*J*_{1}*}k*ref*,L*^{∞}_{x,t}*(L*^{2}_{ξ}*)*

*≤ δ*0*,*

*then the Cauchy problem (1.4) has a global solution in time, and the solution**J(x, t, ξ)*
*decays to zero in the**k · k*ref*,L*^{∞}*x**(L*^{2}_{ξ}*)**norm as**t**→ ∞**.*

*That is, the solution* *F (x, t, ξ)**to (1.1)–(1.3) tends toward the outgoing shock*
*profile**ϕ(x, t, ξ)**in the**k · k*ref*,L*^{∞}_{x}*(L*^{2}_{ξ}*)**norm as**t**→ ∞**.*

* Remark 1.2. Although both the strength of shocks and the magnitude of*
perturbations of the initial data are small, these two parameters are chosen inde-
pendently. When the strength of the shock is sufficiently small, the existence of
the Boltzmann shock profile can be guaranteed and the shocks of the Boltzmann
equation and the Navier-Stokes equation are close enough. Therefore, we can
make use of the compressibility of the Navier-Stokes shock profiles. As for

*δ*0, the smallness assumption is to close the higher order estimate due to the nonlinear- ity of the Boltzmann equation. This is also a common short point of the energy method for viscous conservation laws.

This paper is organized as follows. Basic facts about the Boltzmann equation and the shock profiles are summarized in Section 2. In Section3, the approx- imate solution for the present problem is constructed and the related equations are derived. Section4collects several technical estimates developed to analyze the macroscopic and microscopic equations. The stability analysis is finally shown in Section5. The Chapman-Enskog expansion and the Navier-Stokes equations are briefly described in AppendixA. The properties of the collision operator for hard spheres are put in the AppendixB.

2. P^{RELIMINARY}

* 2.1. Macro-micro decomposition. We first recall several important proper-*
ties of the Boltzmann equation

*F*_{t}*+ ξ · ∇**x**F**= Q(F, F),* *(x, t, ξ)**∈ R*^{3}*× R*^{+}*× R*^{3}*.*

In the present paper we consider the hard sphere as our model, and the collision operator can be written as follows ([11], [12]):

*Q(g, h)**≡*
Z

*R*^{3}

Z

*S*^{2}

h

*g(ξ*^{0}*)h(ξ*_{?}^{0}*)**− g(ξ)h(ξ**?**)*i

*C(**Ω, ξ − ξ**?**)*^{d}Ω^{d}*ξ**?**,*

where

*ξ*^{0}*= ξ + (Ω · (ξ**?**− ξ))Ω,*
*ξ*_{?}^{0}*= ξ**?**− (Ω · (ξ**?**− ξ))Ω,*

*Ω ∈ S*^{2}*.*

The function*C(**Ω, ξ − ξ**?**)*for a hard sphere is

*C(Ω, ξ − ξ**?**)**≡ |Ω · (ξ − ξ**?**)|.*

The local equilibrium distributions are the distributions*F* with*Q(F , F )**=*0, for
which the only solutions are the Maxwellians

*F (ξ)**= ρ*0*ω(ξ*;*u*0*, T*0*),*
where

*ω(ξ*;*u*0*, T*0*)**=* *e*^{−|ξ−u}^{0}^{|}^{2}^{/(}^{2}^{T}^{0}* ^{)}*
p

*(*2

*π T*

_{0}

*)*

^{3}

*.*

The macroscopic density*ρ*_{0}, the velocity*u*_{0}*= (u*01*, u*_{02}*, u*_{03}*)*, and the tempera-
ture *T*_{0} of the local thermal equilibrium state may vary. In the collision process,
mass, momentum, and energy are conserved, i.e., for any distributions*F* and*G*,

Z

*R*^{3}

*Q(F , G)*^{d}*ξ**=*0*,*
Z

*R*^{3}

*ξ**i**Q(F , G)*^{d}*ξ**=*0*,* *i**=*1*,*2*,*3*,*
(2.1)

Z

*R*^{3}*|ξ|*^{2}*Q(F , G)*^{d}*ξ**=*0*.*

Set the collision invariants*χ*_{i}*(ξ)*,*i**=*0*, . . . ,*4, as follows:

*χ*_{0}*(ξ)**=*1*,*
*χ*_{i}*(ξ)**=* *ξ**i*p*− u*0*i*

*T*_{0} *,* *i**=*1*,*2*,*3*,*
*χ*_{4}*(ξ)**=* * _{√}*1

6

*|ξ − u*0*|*^{2}
*T*_{0} *−*3

!
*,*

which are normalized with respect to the Maxwellian state:

Z

*χ*_{i}*χ*_{j}*ω(ξ)*^{d}*ξ**= δ**ij**,* *i, j**=*0*, . . . ,*4*.*

The linearized collision operator

*L(h)**≡ Q(ω, h) + Q(h, ω)*
is self-adjoint and non-positive, i.e.,

*hLg, hi = hg, Lhi,* *hLg, gi ≤*0*.*

Here*h, i*is the inner product on the space*L*^{2}*(R*^{3}*)*with respect to the variable*ξ*:
*hg, hi ≡*

Z
*gh*^{d}*ξ.*

In fact,*χ*_{i}*ω*,*i**=*0*, . . . ,*4, span the null space of*L*. We denote by**P**_{0}the projection
operator on the space spanned by *χ*_{i}*ω*, *i* *=* 0*, . . . ,*4, and by **P**_{1} the orthogonal
projection: **P**_{1}* = I − P*0.

*F*can be decomposed into the macroscopic part

*F*

_{0}and the microscopic part

*F*

_{1}:

*F*_{0}* = P*0

*F*

*=*X4

*i=*0

Z
*χ*_{i}*F*^{d}*ξ*

*χ*_{i}*ω,*
*F*_{1}* = P*1

*F*

*= F − F*0

*.*

It thus follows that

*L(F*_{0}*)**=*0 and *L(F )**= L(F*1*).*

We introduce new norms for*F (x, t, ξ)*, which will be used in the energy estimates:

*kFk**L*^{2}_{ξ}*(x, t)**≡ hF, Fi*^{1}^{/}^{2}*,*
*kFk*_{L}^{∞}_{x,t}_{(L}^{2}

*ξ**)**≡* sup

*(x,t)∈R×R*^{+}

*kFk*_{L}^{2}

*ξ**(x, t).*

We also note that for hard spheres [5] and Grad’s cutoff potentials [11], *L*is
the sum of a multiplication operator and a compact operator*K*:

*Lh(ξ)**= −ν(ξ)h(ξ) + K(h)(ξ).*

The collision frequency*ν(ξ)*has a positive lower bound. As a result,*L*is negative
definite on the microscopic part:

Z

*R*^{3}

*F*_{1}*L(F*_{1}*)*^{d}*ξ <**−ν*0

Z

*R*^{3}

*F*_{1}^{2}^{d}*ξ*

for some positive constant*ν*_{0}. Let^{P}0*≡*ker*(L)*and its orthogonal complement be
denoted by^{P}1. We will write the negative operator restricted to the space^{P}1as

*L*¯*≡ L*

P1 *≤ −ν*0*.*

**Lemma 2.1. For any***g**i**(x, t, ξ)**satisfying***P**_{0}*g**i**≡**0,*

*|hg*1*, Lg*_{2}*i| ≤ −*1

2^{{γhg}^{1}^{, Lg}^{1}^{i + γ}

*−*1*hg*2*, Lg*_{2}*i}*

*for any constant**γ >**0.*

*Proof. By the self-adjoint and non-positive property of**L*. ^{❐}
**Remark 2.2. For more properties of the collision operator***Q*, we refer to
AppendixB.

**2.2. Boltzmann shock profile. Let***ϕ(x−st, ξ)*be a travelling wave solution
of the Boltzmann equation

(2.2) *F**t**+ ξ*1*F**x**= Q(F, F),* *(x, t, ξ)**∈ R × R*^{+}*× R*^{3}*.*
*ϕ*thus satisfies

(2.3) *−sϕ*^{0}*+ ξ*1*ϕ*^{0}*= Q(ϕ, ϕ).*

Let*(ρ, u, T )*denote the macroscopic variables of the travelling wave solution:

*ρ(x**− st) ≡*
Z

*R*^{3}

*ϕ(x**− st, ξ)*^{d}*ξ,*
*m(x**− st) ≡*

Z

*R*^{3}

*ξ*_{1}*ϕ(x**− st, ξ)*^{d}*ξ,* *u**≡* *m*
*ρ* *,*

E*(x**− st) ≡*
Z

*R*^{3}

*|ξ|*^{2}

2 ^{ϕ(x}^{− st, ξ)}^{d}^{ξ,}*m*^{2}

2*ρ* *+ ρT*

!

*≡*^{E}*≡ ρ* *u*^{2}
2 ^{+ e}

!
*.*

Then the states*(ρ*_{±}*, m*_{±}*,*^{E}_{±}*)**≡*lim*x→±∞**(ρ, m,*^{E}*)(x)*satisfy the Rankine-Hugoniot
condition:

*s(ρ*_{−}*− ρ**+**)**= m**−**− m**+**,*

*s(m*_{−}*− m**+**)**= (u**−**m*_{−}*+ p**−**)**− (u**+**m*_{+}*+ p**+**),*
(2.4)

*s(*^{E}_{−}*−*^{E}*+**)**= u**−**(*^{E}_{−}*+ p**−**)**− u**+**(*^{E}_{+}*+ p**+**),*
and the entropy condition

*p*_{−}*− p**+**>*0*.*

Here*p**≡ ρRT*,*R**≡*1. For the existence of the shock profile*ϕ*, see [4].

Denote by*ϕ*T*(x**− st, ξ)*the corresponding local thermal equilibrium distri-
bution:

*ϕ*T*(x**− st, ξ) ≡ ρ(x − st)**e*^{−((ξ}^{1}^{−u(x−st))}^{2}^{+ξ}^{2}^{2}^{+ξ}^{2}^{3}^{)/(}^{2}* ^{T (x−st))}*
p

*(*2

*π T (x*

*− st))*

^{3}

*.*By direct calculations,

*ϕ*Tsatisfies the following lemma.

**Lemma 2.3.**

Z

*R*^{3}*(ϕ**− ϕ*^{T}*)*^{d}*ξ**=*0*,*
Z

*R*^{3}

*ξ**i**√**− u**i*

*T* *(ϕ**− ϕ*^{T}*)*^{d}*ξ**=*0*,* *i**=*1*,*2*,*3*,*
Z

*R*^{3}

*|ξ − u|*^{2}
*T* *−*3

!

*(ϕ**− ϕ*^{T}*)*^{d}*ξ**=*0*.*
*Here for the shock profile**ϕ**of (2.2),**u*1*= u**,**u*2*= u*3*=**0.*

Let*(ρ*NS*, u*NS*,*^{E}NS*)(x**− st)*be the approximate shock profile of the Navier-
Stokes equation obtained by the Chapman-Enskog expansion, which connects the
same end states*(ρ*_{±}*, u*_{±}*,*^{E}_{±}*)*. The corresponding local Maxwellians are denoted
by

*ω*tr*(x**− st, ξ) =* *e*^{−((ξ}^{1}^{−u}^{NS}^{)}^{2}^{+ξ}^{2}^{2}^{+ξ}^{2}^{3}^{)/(}^{2}^{T}^{NS}* ^{)}*
p

*(*2

*π T*NS

*)*

^{3}

*,*

*ϕ*tr*(x**− st, ξ) = ρ*^{NS}*e*^{−((ξ}^{1}^{−u}^{NS}^{)}^{2}^{+ξ}^{2}^{2}^{+ξ}^{3}^{2}^{)/(}^{2}^{T}^{NS}* ^{)}*
p

*(*2

*π T*NS

*)*

^{3}

*.*

We consider a weak shock*ϕ*with strength*ε* *≡ |ρ**−**− ρ**+**| *1. The rate of the
profile*ϕ*converging to the Maxwellian equilibrium states is given in the following
theorem.

**Theorem 2.4. On the Boltzmann shock profile***ϕ(x**− st, ξ)**, there exist* *C*_{1}*,*
*C*_{2}*>**1 and**C*_{3}*∈ (*0*,*1*)**such that*

*|ϕ(x, ξ) − ϕ*^{tr}*(x, ξ)|*

(2.5)

*≤ C*1*ε*^{2}*e*^{−C}^{3}^{ε|x|}*ρ(x)**e*^{−((ξ}^{1}^{−u(x))}^{2}^{+ξ}^{2}^{2}^{+ξ}^{2}^{3}^{)/(}^{2}^{C}^{2}* ^{T (x))}*
p

*(T (x))*

^{3}

*,*

*|∂*^{k}*x**ϕ(x, ξ)|*

(2.6)

*≤ C*1*ε*^{1}^{+k}*e*^{−C}^{3}^{ε|x|}*ρ(x)**e*^{−((ξ}^{1}^{−u(x))}^{2}^{+ξ}^{2}^{2}^{+ξ}^{3}^{2}^{)/(}^{2}^{C}^{2}* ^{T (x))}*
p

*(T (x))*

^{3}

*,*

*k**=*1*, . . . ,**10.*

* Remark 2.5. The above theorem is proved in [4] and the second inequality*
is the consequence of the scalings. We also refer readers to Appendix C of [18],
where the existence of the Boltzmann shock profile with properties (2.5) and (2.6)
has been shown by the weighted energy method.

3. CONSTRUCTION OF THEAPPROXIMATESOLUTION

* 3.1. The macroscopic and microscopic equations. In this section we con-*
struct the approximate solution of (1.1)–(1.3) by superposing two travelling wave
solutions moving in the opposite directions with the same speed.

The state*(ρ*_{+}*, m*_{+}*,*^{E}_{+}*)*is given by (1.2) at*x* *= ∞*. Since (1.3) implies that
the macroscopic velocity and momentum are zero at *x* *=* 0, we first construct a
Boltzmann shock profile with these two given conditions. By solving the Rankine-
Hugoniot condition (2.4) and equation (2.3), there exist*ρ*_{0},^{E}0and a wave speed
*s >* 0 such that *(ρ*_{+}*, m*_{+}*,*^{E}_{+}*)*and *(ρ*_{0}*,*0*,*^{E}_{0}*)*can be connected by a travelling
wave solution *ϕ*_{+}*(x**− st, ξ)* on the whole space *R* *= (−∞, ∞)*. Let *(ρ*_{+}*(x**−*
*st), m*_{+}*(x**− st),*^{E}*+**(x**− st))*denote the corresponding macroscopic variables of
*ϕ*_{+}*(x**− st, ξ)*. According to the construction,

*x→∞*lim*(ρ*_{+}*(x), m*_{+}*(x),*^{E}_{+}*(x))**= (ρ**+**, m*_{+}*,*^{E}_{+}*),*

*x→−∞*lim *(ρ*_{+}*(x), m*_{+}*(x),*^{E}_{+}*(x))**= (ρ*0*,*0*,*^{E}_{0}*).*

We can also construct the other travelling wave solution*ϕ*_{−}*(x**+st, ξ)*in the same
way, satisfying

*x→∞*lim*(ρ*_{−}*(x), m*_{−}*(x),*^{E}_{−}*(x))**= (ρ*0*,*0*,*^{E}0*),*

*x→−∞*lim *(ρ*_{−}*(x), m*_{−}*(x),*^{E}_{−}*(x))**= (ρ**+**,**−m**+**,*^{E}_{+}*).*

In fact, we can construct*ϕ*_{−}*(x, ξ*_{1}*, ξ*_{2}*, ξ*_{3}*)**= ϕ**+**(**−x, −ξ*1*, ξ*_{2}*, ξ*_{3}*)*. It thus follows
that

*ρ*_{+}*(x)**= ρ**−**(−x),*
*m*_{+}*(x)**= −m**−**(−x),*

E*+**(x)**=*^{E}*−**(**−x),*
*T*_{+}*(x)**= T**−**(−x).*

We now choose the approximate solution*ϕ(x, t, ξ)*to be

(3.1) *ϕ(x, t, ξ)**≡ ϕ**+**(x**− s(t + t*0*), ξ)**+ ϕ**−**(x**+ s(t + t*0*), ξ)**− ρ*0*ω(ξ*; 0*, T*0*).*

Here*t*_{0}*≡ ε*^{−}^{3},*ε**≡ |ρ**+**− ρ*0*| *1. It follows from the above construction that
*ϕ(x, t, ξ*_{1}*, ξ*_{2}*, ξ*_{3}*)**= ϕ(−x, t, −ξ*1*, ξ*_{2}*, ξ*_{3}*).*

Let *F* denote a solution of (1.1)–(1.3), extended to the whole space *R* by
setting

*F (x, t, ξ*_{1}*, ξ*_{2}*, ξ*_{3}*)**= F(−x, t, −ξ*1*, ξ*_{2}*, ξ*_{3}*),* for*x <*0*.*
We now write

*F (x, t, ξ)**≡ ϕ(x, t, ξ) + J(x, t, ξ).*

Then by (1.1) and (3.1),*J(x, t, ξ)*satisfies

*J**t**+ ξ*1*J**x**= Q(ϕ + J, ϕ + J) − Q(ϕ, ϕ) − E(ϕ),*
where

*E(ϕ)**≡ ϕ**t**+ ξ*1*ϕ*_{x}*− Q(ϕ, ϕ).*

Also,

*J(x, t, ξ*_{1}*, ξ*_{2}*, ξ*_{3}*)**= J(−x, t, −ξ*1*, ξ*_{2}*, ξ*_{3}*).*

We choose the initial state*J(x,*0*, ξ)*satisfying
Z_{∞}

*−∞*

Z

*R*^{3}

1
*ξ**i*

*|ξ|*^{2}

* J(x,*0*, ξ)*^{d}*ξ*^{d}*x**=*0*,* for*i**=*1*,*2*,*3*.*

Due to the conservation laws for the macroscopic variables, it thus follows that
Z_{∞}

*−∞*

Z

*R*^{3}

1
*ξ**i*

*|ξ|*^{2}

* J(x, t, ξ)*^{d}*ξ*^{d}*x**=*0*,* for*i**=*1*,*2*,*3*.*

We consider the anti-derivative:

*W (x, t, ξ)**≡*
Z*x*

*−∞**J(y, t, ξ)*^{d}*y.*

We thus have

*W**t**+ ξ*1*W**x**=*
Z*x*

*−∞*

*Q(ϕ**+ J, ϕ + J) − Q(ϕ, ϕ) − E(ϕ)*

d*y.*

Let _{}

*ρ*NS^{±}*, u** ^{±}*NS

*,*

^{E}

*NS*

^{±}

*(x**∓ st)*

be approximate shock profiles of the Navier-Stokes equation connecting the same
end states as those of*ϕ*_{±}*(x**∓ st, ξ)*. Define